Precisely coordinated spatio-temporal spiking dynamics have been observed
experimentally in different neuronal systems and are discussed to be an
essential part of computation in the brain. Their dynamical origin,
however, remains unknown.
Here we study the dynamics of neural network models that reveal basic
mechanisms underlying the neurons' precise temporal coordination. Special
emphasis is given on delayed interactions, complicated network
connectivity and computation.
First, we present unstable attractors: invariant periodic orbits with a
positive measure basin that are locally unstable. These occur in networks
of neural oscillators due to delayed interactions. They are enclosed by
basins of attraction of other attractors but are remote from their own
basin volume such that arbitrarily small noise leads to a switching among
attractors. These switching phenomena precisely coordinate spike timing.
They may be useful for artificial neural network computation and operate
in biological neural networks, such as the olfactory system, as well.
Second, we elaborate on the exact dynamics of neural networks exhibiting a
complicated topology. In such networks, an irregular, balanced state
coexists with a synchronous state of regular activity. Using a random
matrix approach, introduced by Wigner in the 1950s to characterize energy
spectra of atomic nuclei, we predict the speed of synchronization in such
networks in dependence of neuron and network properties. We find that the
speed of synchronization is limited by the network connectivity and
remains finite, even if the coupling strengths between neurons becomes
infinite.